On nonexistence of distance regular graphs with the intersection array $\{53,40,28,16;1,4,10,28\}$

We consider $Q$-polynomial graphs of diameter $4.$ Apart from infinite series intersection arrays $\{m(2m+1),(m-1)(2m+1),m^2,$ $m;1,m,m-1,m(2m+1)\}$ there are the following admissible intersection arrays of $Q$-polynomial graphs of diameter $4$ with at most $4096$ vertices: $\{5,4,4,3;1,1,2,2\}$ (odd graph on $9$ vertices), $\{9,8,7,6;1,2,3,4\}$ (folded $9$-cube), $\{36,21,10,3;1,6,15,28\}$ (half $9$-cube), and $\{53,40,28,$ $16;1,4,10,28\}.$ In the paper it is proved that a distance regular graph with an intersection array $\{53,40,28,16;1,4,10,28\}$ does not exist. Bibliogr. 4.